Please help will give brainliest

If surface , S is the part of the sphere, x² + y²+ z² =1 ,lies above the cone, x² + y² = z

a)| ⃗rᵩ × ⃗rθ| = ⟨sin²φ cosθ,sin²φsinθ, sinφcosφ⟩

b)∫∫ z² ds = ₀∫ˣ₀∫ʸcos²φ dφdθ,

S

where x = 2π and y = π/4

A surface in space given in Cartesian coordinates as f(x,y,z) = 0, can be parametrized as a vector function with two parameters,

r(u,v)= ⟨r₁(u,v) , r₂(u,v) , r₃(u,v)⟩ , (u,v)∈R²

We have, S is a part of Sphere , x² + y²+ z² =1 , lies above the cone, x² + y² = z and surface S parametrized by following vector equation ,

r(θ, φ) = ⟨sinφ cosθ , sinφsinθ,cosφ⟩ --(1)

⃗rᵩ = ∂r/∂φ =⟨cos φ cosθ,cos φ sinθ,-sin φ⟩

⃗rθ= ∂r/∂θ=⟨- sinφ sinθ , sinφ cosθ ,0⟩

| ⃗rᵩ × ⃗rθ| =| i j k |

|cosφcosθ cosφ sinθ -sinφ |

|- sinφsinθ sinφ cosθ 0 |

= i( 0 + sinφ cosθsinφ) -j(0- (- sinφsinθ)(-sinφ)) + k(sinφ cosθ cosφcosθ - (-sinφ sinθ ) cosφ sinθ)

= (sin²φcosθ )i + (sin²φsinθ)j + (sinφcosφ(sin²θ+cos²θ))k

= sin²φcosθ )i + (sin²φsinθ)j +(sinφcosφ)k

| ⃗rᵩ× ⃗rθ|=⟨sin²φcosθ ,sin²φsinθ,sinφcosφ⟩

b) from part (a) we get,

x = sinφ cosθ

y = sinφsinθ

z = cosφ

are spherical coordinates of S where, 0≤φ≤π/4 and 0≤ θ≤ 2π.

so, ∫∫ z² ds = ₀∫ˣ₀∫ʸcos²φ dφ dθ ,

S

where x = 2π and y = π/4

So,Surface integeral is equals to ₀∫ˣ₀∫ʸcos²φ dφ dθ.

To learn more about Parametrizing Surfaces, refer:

https://brainly.com/question/29772771

#SPJ4

Complete question:

If Sis the part of the sphere x2 + y2 + x2 = 1 that lies above the cone z= x2 + y2, find the following: a)S can be parametrized by a vector equation

r(θ, φ) = ⟨sinφ cosθ , sinφsinθ , cosφ ⟩ then a)| ⃗rᵩ × ⃗rθ| = ?

b) ∫∫ z² ds = ?

S

The formula to find the slope is m = (y₂ - y₁)/(x₂ - x₁), hence option B is correct and the slope of the line will be -1/2.

It is possible to determine a line's direction and steepness by looking at its slope. Finding the slope between lines inside a coordinate plane can aid in anticipating if the lines are perpendicular, parallel, or none at all without physically using a compass.

As per the data mentioned in the question,

1.

The formula to find the slope of the line between two points is,

m = (y₂ - y₁)/(x₂ - x₁)

2.

The given points are,

(-4, 3) and (12, -5)

So, the slope of the line that connects given points,

m = (-5 - 3)/(12+4)

m = -8/16

m = -1/2

To know more about Slope:

https://brainly.com/question/3605446

#SPJ1

Answer:

C) x = 9

Step-by-step explanation:

[tex]\frac{12}{27}[/tex] = [tex]\frac{4}{x}[/tex]

[tex]\frac{12}{27}[/tex] ÷ [tex]\frac{3}{3}[/tex] = [tex]\frac{4}{9}[/tex]  This is the fraction simplified

[tex]\frac{4}{9}[/tex] = [tex]\frac{4}{x}[/tex]  Simplified, I think that it is easier to see that x = 9

The required null hypothesis of the given observation is μ1 = μ2 = μ3.

The null hypothesis is a typical mathematical theory that asserts that there is no statistical relationship and significance between two sets of observed data and measured phenomena for each set of specified, single observable variables.

The null hypothesis can be evaluated to determine whether or not there is a relationship between two measured phenomena, which makes it valuable.

It can let the user know if the outcomes are the product of random chance or deliberate manipulation of a phenomenon.

The null hypothesis, often known as "H-nought," "H-null," or "H-zero," is written as H0 to distinguish it from other hypotheses.

The null hypothesis of the given observation:

μ1 = μ2 = μ3

Therefore, the required null hypothesis of the given observation is μ1 = μ2 = μ3.

Know more about the null hypothesis here:

https://brainly.com/question/13045159

#SPJ4

Complete question:
To test whether or not there is a difference between treatments A, B, and C, a sample of 12 observations has been randomly assigned to the 3 treatments. You are given the results below.

Treatment

Observations

A

20

30

25

33

B

22

26

20

28

C

40

30

28

22

The null hypothesis for this ANOVA problem is?

Answer:

Step-by-step explanation:

The value of y is unclear given the information provided. However, if BC intersects AD at E, then it stands to reason that y would be equal to the value of x. This is because, if BC intersects AD at E, then the two lines are parallel and therefore have the same value for y. Calculating the internal angles of triangle DEF, we can find the length of BW, CW and CE. Since BD=87 and WE=38, the remaining angle must be 55°. Therefore, using trigonometry and the known side lengths, we can calculate BW = 87 sin 55° = 77.5  , CW = 87cos55° = 48.5  and CE = 38sin55° = 33.2 . As you can see then, these are all easily calculable from the given data. In angle DEF, BD = 87 and WE = 38. Therefore, BW = 87 - 38 = 49, CW = 49 - 38 = 11, and CE = 11 - 1 = 10.

a) The linear equation for R(t) is given as follows: R(t) = -2.1t + 16.

b) The helium reserves in 2015 will be of: 5.5 billion.

c) The reserve will be depleted in the year of: 2018.

The linear function in this problem has the definition in slope-intercept format presented as follows:

R(t) = mt + b.

In which:

Considering the measures in billions, the parameters are given as follows:

m = -2.1, b = 16.

Then the equation is defined as follows:

H(t) = -2.1t + 16.

2015 is five years after 2010, hence the estimate is calculated as follows:

H(5) = -2.1(5) + 16 = 5.5 billion.

The reserve will be depleted in the year t + 2010 + 1, when R(t) = 0, hence:

-2.1t + 16 = 0

2.1t = 16

t = 16/2.1

t = 7.61 years.

Hence the reserves will be depleted during the year of 2018.

More can be learned about linear functions at https://brainly.com/question/24808124

#SPJ1

Let A be the amount of money invested at a 12% annual interest rate and B be the amount of money invested at a 8% annual interest rate.

We know that A = B + 1500, and the total interest earned by the two investments is $1880.

The interest earned by the investment at a 12% annual interest rate is 0.12 * A = 0.12A

The interest earned by the investment at a 8% annual interest rate is 0.08 * B = 0.08B

Therefore, we can write the following equation to represent the situation:

0.12A + 0.08B = 1880

Since A = B + 1500, we can substitute this into the equation to get:

0.12(B + 1500) + 0.08B = 1880

Solving for B, we get:

0.12B + 180 + 0.08B = 1880

Combining like terms, we get:

0.2B + 180 = 1880

Subtracting 180 from both sides, we get:

0.2B = 1700

Dividing both sides by 0.2, we get:

B = 8500

Since A = B + 1500, we can substitute this value into the equation to find the value of A:

A = 8500 + 1500 = 10000

Therefore, the investor invested $8,500 at an 8% annual interest rate and $10,000 at a 12% annual interest rate.

Answer:

Account A (12%)= $10,000

Account B (8% stock fund) = $8,500

Step-by-step explanation:

Annual Interest Formula

[tex]\large \text{$ \sf I=P\left(1+r\right)^{t} -P$}[/tex]

where:

Account A:

[tex]\implies \sf Interest=(x+1500) (1+0.12)^1-(x+1500)[/tex]

[tex]\implies \sf Interest=(x+1500) (1+0.12)-(x+1500)[/tex]

[tex]\implies \sf Interest=(x+1500)+0.12(x+1500)-(x+1500)[/tex]

[tex]\implies \sf Interest=0.12(x+1500)[/tex]

[tex]\implies \sf Interest=0.12x+180[/tex]

Account B (stock fund):

[tex]\implies \sf Interest=x (1+0.08)^1-x[/tex]

[tex]\implies \sf Interest=x (1+0.08)-x[/tex]

[tex]\implies \sf Interest=x+0.08x-x[/tex]

[tex]\implies \sf Interest=0.08x[/tex]

If the total interest from the two investments was $1880:

[tex]\implies \sf 0.12x+180+0.08x=1880[/tex]

[tex]\implies \sf 0.2x+180=1880[/tex]

[tex]\implies \sf 0.2x=1700[/tex]

[tex]\implies \sf x=8500[/tex]

Therefore, the money invested in each of the two accounts is:

The parameters of the exponential function in this problem are given as follows:

The standard format of an exponential function is given as follows:

y = c(1 - a)^t.

This is the case for a decaying exponential function, and the meaning of each parameter is given as follows:

In this problem, the function is given as follows:

f(x) = 1.1(0.44)^x.

Hence the values for the parameters are given as follows:

Then the percent of change is of -56%, as it is a decaying exponential function with a = 0.56.

More can be learned about exponential functions at https://brainly.com/question/25537936

#SPJ1

Answer:

.

Step-by-step explanation:

sorry i hit my daily limit and i need more answers

[tex]10+2a < 4[/tex]

Simplify:

[tex]2a+10 < 4[/tex]

Subtract 10 from both sides:

[tex]2a+10-10 < 4-10[/tex]

[tex]2a < -6[/tex]

Divide both sides by 2:

[tex]\dfrac{2a}{2} < \dfrac{-6}{2}[/tex]

[tex]a < -3[/tex]

[tex]\fbox{Second Option}[/tex]

Given the graph of f(x), the range of f⁻¹(x) is; (−∞, 2) ∪ (2, ∞)

The range of a function is defined as the set that is composed by all the output values on a function. Thus, on the graph, the range of a function is composed by the values of y of the function.

For the inverse function, the input and the output are exchanged, and as such given the graph of a function, we can say that the range of the inverse function is given by the domain of the initial function, which is, the values of x of the graph of the original function.

From the description of the rational function, we are told that the asymptote is given as: x = 2

Thus, the domain of the graphed function, would be the same as the range of the inverse function,  will be given by the interval:

(−∞, 2) ∪ (2, ∞).

Read more about range of function at; https://brainly.com/question/7954282

#SPJ1

By applying vertical compression by a factor of 3, then vertical reflection and finally horizontal translation by 1 unit to the right we can transform into required graph.

A transformation is a mathematical operation that changes the position, size, or shape of a geometric object. In other words, it is a way to manipulate the object by moving, scaling, rotating, or reflecting it.

Reflection is a transformation that flips an object over a mirror line, or line of reflection. It creates a mirror image of the object on the other side of the line.

In mathematics, reflection is often represented by a matrix called the reflection matrix. This matrix encodes the rules for reflecting points in a particular direction.

To transform the graph of y = x^2 into y = -3(x+1)^2, we can apply the following transformations:

Vertical stretch or compression by a factor of 3: This transformation stretches or compresses the graph vertically by a factor of 3. To apply this transformation, we need to multiply the y-values of the points on the graph by 3.

Vertical reflection: This transformation reflects the graph across the x-axis. To apply this transformation, we need to multiply the y-values of the points on the graph by -1.

Horizontal translation by 1 unit to the right: This transformation shifts the graph 1 unit to the right. To apply this transformation, we need to add 1 to the x-values of the points on the graph.

Applying these transformations to the graph of y = x^2, we get the graph of y = -3(x+1)^2.

To learn more about transformation visit:

https://brainly.com/question/11352944

#SPJ1

According to the given statement the positive exponent is [tex]\frac{16}{625}[/tex].

Exponents are a method of expressing enormous magnitudes in terms of their respective powers. The amount of times some number has already been multiplied in itself is the exponent, so to speak. For instance, the result of multiplying the number 6 on it's own four times is 6 6 6 6. This may be expressed as 64. Here, the exponent and base are 4 and 6, respectively.

The exponent is the exact quantity of times that base would be compounded on its own. As a result, if two powers have the same foundation, they can be multiplied. Whenever two powers are multiplied, exponents are added. If necessary, we can also divide the abilities.

= [tex]( \frac{2}{5} )^{4}[/tex]

= [tex]\frac{2}{5} \times\frac{2}{5} \times\frac{2}{5} \times\frac{2}{5} \\\\\frac{16}{625}[/tex]

Hence, the required exponent is [tex]\frac{16}{625}[/tex].

To know more about exponent visit:

https://brainly.com/question/15993626

#SPJ1

The estimated distance traveled by the car is 73.5 mi.

To use the Midpoint Rule, we need to divide the time interval into subintervals and evaluate the average velocity over each subinterval.

The time interval is from 0 seconds to 100 seconds, so we will divide this interval into 5 subintervals of length 20 seconds each. The midpoint of each subinterval is the average time, which we can use to estimate the average velocity over that subinterval.

The midpoints of the subintervals are:

Using the data from the table, we can calculate the average velocity over each subinterval:

We can now use the Midpoint Rule to estimate the distance traveled by the car:

distance = (20/6) * (17 + 53 + 53 + 49.5 + 47)

              = (20/6) * 220.5

              = <<20/6*220.5=73.5>>73.5 mi

So, the estimated distance traveled by the car is 73.5 mi.

Learn More about Midpoint Rule:

https://brainly.com/question/2257264

#SPJ4

Consequently, the rectangular garden's length is [tex]x^{2} +2x+2[/tex]

option D is correct

Area is the entire amount of space occupied by a flat (2-D) surface or an object's shape. Surface area refers to an open surface or the perimeter of a three-dimensional object, whereas plane region or plane area refers to a shape or planar lamina.

given

the width(w) of the rectangular garden= [tex]x[/tex][tex]-2[/tex]

area of the rectangular garden = [tex]x^{3} -2x-4[/tex]

The area of a rectangle is,

area = length × width(w) ,

If l is the rectangle's length,

w is the width of the rectangle,

By substituting values,

[tex]x^{3}-2x-4[/tex] = ([tex]x-2[/tex]) *  length ( l )

[tex]l=\frac{x^{3}-2x-4 }{x-2}[/tex] = [tex]x^{2} +2x+2[/tex]   (by long division method)

Consequently, the rectangular garden's length is [tex]x^{2} +2x+2[/tex]

option D is correct

To know more about area visit:-

https://brainly.com/question/27683633

#SPJ1

The true statements about the graph are 1, 2, and 3.

A line of the best fit is a straight line that reduces the distance between it and some data. The line of best fit is used to represent a relationship in a scatter plot with numerous data points.

Given:

The temperature above 52° F affects the sale of hot chocolate at her shop,

As you can see from the graph, the y-axis shows the orders for hot chocolate on a day when the outside temperature was 52° F.

From the graph,

at x = 0, y = 160

Thus, Sheryl gets 160 orders for hot chocolate on a day when the outside temperature is 52° F

At x = 0, y = 160 and at x = 16, y = 128

So the difference is 160-128 / 16 - 0 = 2

Thus, Sheryl gets 2 fewer orders for each degree increase above 52° F in the outside temperature.

To know more about the line of fits:

brainly.com/question/22992800

#SPJ1

Answer:

-9

Step-by-step explanation:

Given function:

[tex]f(x)=-9x+4[/tex]

Given x-values:

Calculate the value of the function for the two given values of x:

[tex]\begin{aligned}\implies f(x_1)&=-9(-5)+4\\&=45+4\\&=49\end{aligned}[/tex]

[tex]\begin{aligned}\implies f(x_2)&=-9(0)+4\\&=0+4\\&=4\end{aligned}[/tex]

[tex]\boxed{\begin{minipage}{6.3 cm}\underline{Average rate of change of function $f(x)$}\\\\$\dfrac{f(b)-f(a)}{b-a}$\\\\over the interval $a \leq x \leq b$\\\end{minipage}}[/tex]

As -5 < 0:

Therefore:

[tex]\begin{aligned} \implies \textsf{Average rate of change}&=\dfrac{f(x_2)-f(x_1)}{x_2-x_1}\\\\&=\dfrac{4-49}{0-(-5)}\\\\&=\dfrac{-45}{5}\\\\&=-9\end{aligned}[/tex]

The absolute value equation represented in the graph is

|x + 3|  

Without taking direction into account, absolute value describes how far away from zero a certain number is on the number line.

A number can never have a negative absolute value.

The equation is written in the form below

|x|

The a transformation which involves a translation of 3 units to the left took place.  the transformation rule for 3 units to the left is addition of 3, hence we have the equation

|x + 3|

We can therefore conclude that the absolute value equation of the function is |x + 3|

Learn more absolute values here:

https://brainly.com/question/29532815

#SPJ1

Properties of log:

Evaluate given using the properties above:

Q1

Q2

Q3

Answer:

[tex]\textsf{1.} \quad 5 \ln x + 5 \ln y[/tex]

[tex]\textsf{2.} \quad \dfrac{1}{7} \ln t[/tex]

[tex]\textsf{3.} \quad \ln x - \dfrac{5}{2}\ln y \;\; \;\;\textsf{or} \;\;\;\; \ln x - 5\ln y[/tex]

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{6 cm}\underline{Natural log laws}\\\\Product law: \;$\ln xy=\ln x + \ln y$\\\\Quotient law: $\ln \left(\dfrac{x}{y}\right) = \ln x - \ln y$\\\\Power law: \;\;\;\;$\ln x^n=n \ln x$\\\end{minipage}}[/tex]

Apply the power law followed by the product law:

[tex]\begin{aligned}\ln (xy)^5 & = 5 \ln (xy)\\ & = 5\left( \ln x + \ln y \right) \\ & = 5 \ln x + 5 \ln y\end{aligned}[/tex]

[tex]\textsf{Apply the exponent rule} \quad \sqrt[n]{a}=a^{\frac{1}{n}}[/tex]

then apply the power law:

[tex]\begin{aligned}\ln \sqrt[7]{t} & = \ln t^{\frac{1}{7}} \\ & = \dfrac{1}{7} \ln t\end{aligned}[/tex]

It is not completely clear where the square root sign begins and ends, so I have provided answers for both permutations:

[tex]\begin{aligned}\ln \left(\sqrt{\dfrac{x^2}{y^5}}\right) & = \ln \left(\dfrac{\sqrt{x^2}}{\sqrt{y^5}}\right) \\\\& = \ln \left(\dfrac{x}{y^{\frac{5}{2}}}\right) \\\\&=\ln x - \ln y^{\frac{5}{2}}\\\\&=\ln x - \dfrac{5}{2}\ln y\end{aligned}[/tex]

[tex]\begin{aligned}\ln \left(\dfrac{\sqrt{x^2}}{y^5}\right)& = \ln \left(\dfrac{x}{y^5}\right) \\\\&=\ln x - \ln y^5\\\\&=\ln x - 5\ln y\end{aligned}[/tex]

Answer:

This is a statement. Not a question.

Step-by-step explanation:

The maximum dimension of a rectangular microchip she can use is 12 mm and 24 mm.

A rectangular microchip, commonly known as an integrated circuit or chip, resembles a flat rectangle approximately small in size with a slew of wires designated by pins protruding from it.

The maximum dimension of a rectangular microchip can be determined by finding its length and width.

From the information given:

The surface area of the rectangle can be expressed by using the formula:

S = 2(lw + lh + wh)

where;

The surface area then becomes:

864 = 2(2x² + 8x + 4x)

Divide both sides by 2, and we have:

432 = 2x² + 12x

We can now have a quadratic expression as:

2x² + 12x - 432 = 0

x² + 6x - 216 = 0

using the  factorization method;

(x + 18) (x - 12) = 0

x = - 18 or x = 12;

Since x cannot be negative, then x = 12. Therefore, width = 12 mm, and the length = 2(12) = 24 mm. Therefore, we can conclude that the maximum dimension of the microchip she can use is 12 mm and 24 mm.

Learn more about finding the maximum dimension of a rectangular shape here:

https://brainly.com/question/19819849

#SPJ1

Answer:

I went with B. 9 mm and 18 mm   PLATO

Step-by-step explanation:  mad maths skills

The values of v and w that will make ΔPQR congruent to ΔKJI are:

v = 12; w = 20.

Based on the CPCTC, every corresponding sides and corresponding angles of two congruent triangles are equal to each other.

Therefore, given that ΔPQR ≅ ΔKJI

PR = KI, and QR = JI

Substitute the values:

5v = 6v - 12

5v - 6v = -12

-v = -12

v = 12

w - 1 = 3w - 41

w - 3w = 1 - 41

-2w = -40

w = -40/-2

w = 20

Learn more about congruent triangles on:

https://brainly.com/question/1675117

#SPJ1

Answer:

y = 1/4x + 1

Step-by-step explanation:

The correct statement regarding the inverse function of f(x) = 4x^4 is given as follows:

[tex]f^{-1}(x) = \pm \left(\frac{x}{4}\right)^{\frac{1}{4}}[/tex]; f^(-1)(x) is not a function.

The function in this problem is defined as follows:

f(x) = 4x^4.

To obtain the inverse of a function y = f(x), first the variables y and x are exchanged, as follows:

x = 4y^4.

Isolating the variable y, we have that:

y^4 = (x/4).

The inverse operation of the fourth power is the fourth root, hence:

[tex]y = \pm \sqrt[4]{\frac{x}{4}}[/tex]

[tex]f^{-1}(x) = \pm \sqrt[4]{\frac{x}{4}}[/tex]

[tex]f^{-1}(x) = \pm \left(\frac{x}{4}\right)^{\frac{1}{4}}[/tex]

The plus/minus symbol means that for each input of x, the inverse function gives two outputs, meaning that there are multiple outputs mapped to each input, and thus the inverse is not a function.

This means that the second statement is correct.

More can be learned about inverse functions at https://brainly.com/question/3831584

#SPJ1

Step-by-step explanation:

To solve the equation -16(d + 1) = -21 for d, we can first use the distributive property to expand the expression -16(d + 1):

-16(d + 1) = -16d - 16

Then, we can set the two sides of the equation equal to each other and solve for d:

-16d - 16 = -21

-16d = -5

d = 5/16

Therefore, the value of d that satisfies the equation -16(d + 1) = -21 is d = 5/16.

Yes, I agree that this equivalent expression has been simplified.

Reorder and gather like terms: (8.4x +6.3 x) +12.6

Collect coefficients for the like terms: (8.4 +6.3)  ×  x +12.6

Calculate the sum or difference: 14.7 x +12.6

Answer : 14.7 x  + 12.6

What exactly is equivalent expression, and how do we recognize it?

Equivalent expressions are expressions that work the same even though they look different. If two algebraic expressions are equivalent, then the two expressions have the same value when we plug in the same value(s) for the variable(s).

To learn more about coefficients refer to:

https://brainly.com/question/1038771

#SPJ1

First, multiply [tex]18*5 = 90[/tex]

Then, divide [tex]90 / 6 = 15[/tex]

Therefore, each group has 15 students.

Answer:

124 and 228

Step-by-step explanation:

Two numbers will be presented as x and y

x + y = 352

x - y = 104

x + y + x - y = 352 + 104

2x = 456

Divide both side by 2

x = 228

x + y = 352

y = 352 - x

y = 352 - 228

y = 124